Microbundle
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− | == | + | {{Authors|Diarmuid Crowley|Matthias Kreck}} |
+ | ==Definition== | ||
<wikitex>; | <wikitex>; | ||
− | The concept of | + | The concept of a '''microbundle''' of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[Wikipedia:Topological manifold|topological manifold]]. Later Kister \cite{Kister1964}, and independently Mazur, showed that every microbundle uniquely determines a topological $\Rr^n$-bundle; i.e. a fibre bundle with structure group the homeomorphisms of $\Rr^n$ fixing $0$. |
− | + | ||
{{beginthm|Definition|{{cite|Milnor1964}} }} | {{beginthm|Definition|{{cite|Milnor1964}} }} | ||
− | An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ | + | Let $B$ be a topological space. An '''$n$-dimensional microbundle''' over $B$ is a quadruple $(E,B,i,j)$ |
− | #$j\circ i=\id_B$ | + | where $E$ is a space, $i$ and $j$ are maps fitting into the following diagram |
− | # | + | $$B\xrightarrow{i} E\xrightarrow{j} B$$ |
+ | and the following conditions hold: | ||
+ | #$j\circ i=\id_B$. | ||
+ | #For all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$ | ||
which makes the following diagram commute: | which makes the following diagram commute: | ||
− | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | + | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}. $$ |
+ | The space $E$ is called the '''total space''' of the bundle and $B$ the '''base space'''. | ||
+ | |||
+ | Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are '''isomorphic''' if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute: | ||
+ | $$ | ||
+ | \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ | ||
+ | B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ | ||
+ | & V_2 \ar[ur]_{j_2|_{V_2}} } | ||
+ | $$ | ||
{{endthm}} | {{endthm}} | ||
+ | </wikitex> | ||
+ | |||
+ | == Examples == | ||
+ | <wikitex>; | ||
+ | An important example of a microbundle is the '''tangent microbundle''' of a topological (or similarly $PL$) manifold $M$. | ||
+ | Let | ||
+ | $$\Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$$ | ||
+ | be the diagonal map for $M$. | ||
{{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | {{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | ||
− | Let $M$ be topological $n$-manifold, | + | Let $M$ be topological (or PL) $n$-manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then |
$$ (M \times M, M, \Delta_M, p_1) $$ | $$ (M \times M, M, \Delta_M, p_1) $$ | ||
− | is an $n$-dimensional microbundle. | + | is an $n$-dimensional microbundle, the '''tangent microbundle''' $\tau_M$ of $M$. |
{{endrem}} | {{endrem}} | ||
− | {{beginrem| | + | |
− | is | + | {{beginrem|Remark}} |
+ | An atlas of $M$ gives a product atlas of $M \times M$ which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold $M$ in $M \times M$ is isomorphic to its tangent bundle. | ||
{{endrem}} | {{endrem}} | ||
− | + | Another important example of a microbundle is the micro-bundle defined by a topological topological $\Rr^n$-bundle. | |
− | + | ||
− | + | {{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. Then the quadruple | |
− | + | $$(E, B, s, \pi)$$ | |
− | + | is an $n$-dimensional microbundle. | |
− | B\ | + | {{endrem}} |
− | + | </wikitex> | |
− | + | == The Kister-Mazur Theorem == | |
− | + | <wikitex>; | |
− | + | A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur. | |
− | + | ||
− | {{beginthm|Theorem|\cite{Kister1964|Theorem 2} }} | + | {{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} |
− | Let $(E, B, i, j)$ be an $n$-dimensional microbundle. | + | Let $(E, B, i, j)$ be an $n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$. |
+ | Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that the following hold. | ||
# $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. | # $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. | ||
− | # | + | # $(E_1, B, i, j|_{E_1})$ is a microbundle and the the inclusion $E_1 \to E$ is a microbundle isomorphism. |
− | # If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 | + | # If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1, B, i, j|_{E_1}) \cong (E_2, B, i, j|_{E_2})$. |
{{endthm}} | {{endthm}} | ||
+ | {{beginrem|Remark}} | ||
+ | Microbundle theory is an important part of the work by Kirby and Siebenmann {{cite|Kirby&Siebenmann1977}} on smooth structures and $PL$-structures on higher dimensional topological manifolds. | ||
+ | {{endrem}} | ||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
+ | == External links == | ||
+ | * The Wikipedia page about [[Wikipedia:Microbundle|microbundles]]. | ||
+ | |||
+ | [[Category:Definitions]] |
Latest revision as of 14:20, 16 May 2013
An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:20, 16 May 2013 and the changes since publication. |
The users responsible for this page are: Diarmuid Crowley, Matthias Kreck. No other users may edit this page at present. |
Contents |
1 Definition
The concept of a microbundle of dimension was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later Kister [Kister1964], and independently Mazur, showed that every microbundle uniquely determines a topological -bundle; i.e. a fibre bundle with structure group the homeomorphisms of fixing .
Definition 1.1 [Milnor1964] .
Let be a topological space. An -dimensional microbundle over is a quadrupleTex syntax error
where is a space, and are maps fitting into the following diagram
Tex syntax error
and the following conditions hold:
Tex syntax error
.- For all
Tex syntax error
there exist open neigbourhoodTex syntax error
, an open neighbourhoodTex syntax error
ofTex syntax error
and a homeomorphismTex syntax error
which makes the following diagram commute:
The space is called the total space of the bundle and the base space.
Two microbundlesTex syntax error, over the same space are isomorphic if there exist neighbourhoods
Tex syntax errorof
Tex syntax errorand
Tex syntax errorof
Tex syntax errorand a homeomorphism
Tex syntax errormaking the following diagram commute:
2 Examples
An important example of a microbundle is the tangent microbundle of a topological (or similarly ) manifold . Let
be the diagonal map for .
Example 2.1 [Milnor1964, Lemma 2.1].
Let be topological (or PL) -manifold, and letTex syntax errorbe the projection onto the first factor. Then
Tex syntax error
Tex syntax errorof .
Remark 2.2. An atlas of gives a product atlas of which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold in is isomorphic to its tangent bundle.
Another important example of a microbundle is the micro-bundle defined by a topological topological -bundle.
Tex syntax errorbe a topological -bundle with zero section
Tex syntax error. Then the quadruple
Tex syntax error
is an -dimensional microbundle.
3 The Kister-Mazur Theorem
A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur.
Theorem 3.1 [Kister1964, Theorem 2].
LetTex syntax errorbe an -dimensional microbundle over a locally finite, finite dimensional simplicial complex . Then there is a neighbourhood of
Tex syntax error,
Tex syntax errorsuch that the following hold.
- is the total space of a topological -bundle over .
- is a microbundle and the the inclusion
Tex syntax error
is a microbundle isomorphism. - If
Tex syntax error
is any other such neighbourhood ofTex syntax error
then there is a -bundle isomorphism .
Remark 3.2. Microbundle theory is an important part of the work by Kirby and Siebenmann [Kirby&Siebenmann1977] on smooth structures and -structures on higher dimensional topological manifolds.
4 References
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
- [Kister1964] J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR0180986 (31 #5216) Zbl 0131.20602
- [Milnor1964] J. Milnor, Microbundles. I, Topology 3 (1964), no.suppl. 1, 53–80. MR0161346 (28 #4553b) Zbl 0124.38404
5 External links
- The Wikipedia page about microbundles.